Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x^2)/(x^4 - 1)
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1
Factor the denominator as a difference of squares: .
Further factor as .
Now the denominator is .
Set up the partial fraction decomposition: .
Multiply through by the common denominator and equate coefficients to solve for , , , and .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations like addition, subtraction, and decomposition. In this case, the expression (x^2)/(x^4 - 1) is a rational expression that requires analysis of its components to simplify or decompose it.
Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the original fraction into simpler parts that can be more easily manipulated or integrated.
Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential in partial fraction decomposition, as the first step is to factor the denominator completely. For the expression (x^2)/(x^4 - 1), recognizing that x^4 - 1 can be factored into (x^2 - 1)(x^2 + 1) is key to finding the appropriate partial fractions.